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%TCIDATA{Created=Wed Apr 02 16:34:41 2003}
%TCIDATA{LastRevised=Thu Apr 10 14:30:17 2003}

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\begin{document}


Definitions: 
\begin{eqnarray*}
\alpha _i^{\prime } &=&-I\gamma _0\gamma _i \\
\Sigma _i^{\prime } &=&\sum_{j<k}-I\varepsilon ^{ijk}\gamma _j\gamma _k
\end{eqnarray*}

New Heavy Quark Fermilab Action

\[
S=S_{01}+S_{02}+S_{03}+S_{04}+S_{05}+S_B+S_E+\sum_{i=1..10}S_i+S_{t1}+S_{t2} 
\]

\begin{itemize}
\item  $S_{01}=1$

\item  $S_{02}=\frac 1{2m_0}[\gamma _0U_0-\gamma _0U_{-0}]$

\item  $S_{03}=-\frac{r_t}{2m_0}[U_0-2+U_{-0}]$

\item  $S_{04}=\frac \zeta {2m_0}\sum_i\gamma _i[U_i-U_{-i}]$

\item  $S_{05}=-\frac{r_s\zeta }{2m_0}\sum_i[U_i-2+U_{-i}]$

\item  $S_B=-\frac{c_Bc_{SW}\zeta }{2m_0}\sum_i\Sigma _i^{\prime }B_i$

\item  $S_E=-\frac{c_Ec_{SW}\zeta }{2m_0}\sum_i\alpha _i^{\prime }E_i$

\item  $S_1=\frac{c_1}{2m_0}\sum_i\sum_j[\gamma
_i(U_i-U_{-i})(U_j-2+U_{-j})+\gamma _j(U_i-2+U_{-i})(U_j-U_{-j})]$

\item  $S_2=\frac{c_2}{2m_0}\sum_i\gamma _i(U_i-U_{-i})(U_i-2+U_{-i})$

\item  $S_3=\frac{2c_3}{2m_0}\sum_i\sum_j(U_i-2+U_{-i})(U_j-2+U_{-j})$

\item  $S_4=\frac{2c_4}{2m_0}\sum_i(U_i-2+U_{-i})(U_i-2+U_{-i})$

\item  $S_5=\frac{c_5}{2m_0}\sum_i\sum_j[\gamma _i\Sigma _j^{\prime
}(U_i-U_{-i})B_j+\Sigma _i^{\prime }\gamma _jB_i(U_j-U_{-j})]$

\item  $S_{10}=\frac{2c_{10}}{2m_0}\sum_i\sum_{j\neq i}[\Sigma _i^{\prime
}B_i(U_j-2+U_{-j})+\Sigma _j^{\prime }(U_i-2+U_{-i})B_j]$
\end{itemize}

From which we conclude:

\begin{itemize}
\item  $id=\stackrel{\text{read note}}{\overbrace{1+\frac{2r_t}{2m_0}+\frac{%
2r_s}{2m_0}}}+\stackrel{S_3}{\overbrace{\frac{2c_3}{2m_0}(3*2+9*4)}}+%
\stackrel{S_4}{\overbrace{\frac{2c_4}{2m_0}(3*2+3*4)}}$

The first term is set to 1 and $\frac 1{2m_0}=k_t$

\item  $u_0=\frac 1{2m_0}\gamma _0-\frac{r_t}{2m_0}1$

\item  $d_0=-\frac 1{2m_0}\gamma _0-\frac{r_t}{2m_0}1$

\item  $u_i=\frac \zeta {2m_0}\gamma _i-\frac{r_s\zeta }{2m_0}1+\stackrel{S_1%
}{\overbrace{\frac{-12c_1}{2m_0}\gamma _i}}+\stackrel{S_2}{\overbrace{\frac{%
-2c_2}{2m_0}\gamma _i}}+\stackrel{S_3}{\overbrace{\frac{-12*2c_3}{2m_0}1}}+%
\stackrel{S_4}{\overbrace{\frac{-4*2c_4}{2m_0}1}}$

\item  $d_i=-\frac \zeta {2m_0}\gamma _i-\frac{r_s\zeta }{2m_0}1+\stackrel{%
S_1}{\overbrace{\frac{+12c_1}{2m_0}\gamma _i}}+\stackrel{S_2}{\overbrace{%
\frac{+2c_2}{2m_0}\gamma _i}}+\stackrel{S_3}{\overbrace{\frac{-12*2c_3}{2m_0}%
1}}+\stackrel{S_4}{\overbrace{\frac{-4*2c_4}{2m_0}1}}$

\item  $e_i=-\frac{c_Ec_{SW}\zeta }{2m_0}\alpha _i^{\prime }$

\item  $b_i=-\frac{c_Bc_{SW}\zeta }{2m_0}\Sigma _i^{\prime }+\stackrel{S_{10}%
}{\overbrace{\frac{2(-4)2c_{10}}{2m_0}\Sigma _i^{\prime }}}$

\item  $u_iu_j=\frac{c_1}{2m_0}(\gamma _i+\gamma _j)+\delta _{ij}\frac{c_2}{%
2m_0}\gamma _i+\frac{2c_3}{2m_0}+\delta _{ij}\frac{2c_4}{2m_0}$

\item  $u_id_j=\frac{c_1}{2m_0}(\gamma _i-\gamma _j)+\delta _{ij}\frac{c_2}{%
2m_0}\gamma _i+\frac{2c_3}{2m_0}+\delta _{ij}\frac{2c_4}{2m_0}$

\item  $d_iu_j=-\frac{c_1}{2m_0}(\gamma _i-\gamma _j)-\delta _{ij}\frac{c_2}{%
2m_0}\gamma _i+\frac{2c_3}{2m_0}+\delta _{ij}\frac{2c_4}{2m_0}$

\item  $d_id_j=-\frac{c_1}{2m_0}(\gamma _i+\gamma _j)-\delta _{ij}\frac{c_2}{%
2m_0}\gamma _i+\frac{2c_3}{2m_0}+\delta _{ij}\frac{2c_4}{2m_0}$

\item  $b_iu_j=\frac{c_5}{2m_0}\Sigma _i^{\prime }\gamma _j+(1-\delta _{ij})%
\frac{2c_{10}}{2m_0}\Sigma _i^{\prime }$

\item  $b_id_j=-\frac{c_5}{2m_0}\Sigma _i^{\prime }\gamma _j+(1-\delta _{ij})%
\frac{2c_{10}}{2m_0}\Sigma _i^{\prime }$

\item  $u_ib_j=\frac{c_5}{2m_0}\gamma _i\Sigma _j^{\prime }+(1-\delta _{ij})%
\frac{2c_{10}}{2m_0}\Sigma _j^{\prime }$

\item  $d_ib_j=-\frac{c_5}{2m_0}\gamma _i\Sigma _j^{\prime }+(1-\delta _{ij})%
\frac{2c_{10}}{2m_0}\Sigma _j^{\prime }$
\end{itemize}

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